Advances in signal acquisition hardware and software have significantly improved coherent processing of a variety of signal modalities. This enables increasingly accurate distance measurements using technologies such as ultrasonic sensing, and millimeter wave radar. With non-penetrating coherent signals, it is possible to measure and reconstruct the depth in a scene and form a depth map.
Active Sensor Arrays
Typical coherent active arrays include transmitting and receiving components. These, depending on the sensing modality and the available hardware, can be separate physical devices or the same transducer. Each transmitter transmits a pulse, which is reflected from an objects in the scene and received by the receivers. A coherent receiver receives the waveform of the reflected pulse, which is processed to recover the desired information from the scene. This is in contrast to incoherent receivers, such as visible-light sensors, which can only acquire a time-averaged energy of the received signal.
The ability to acquire the reflected waveform allows coherent arrays to measure the time-of-flight of the transmitted pulse from the instance it is transmitted until the reflected pulse is received. It is thus possible to estimate the distance and the position of the reflectors in the scene, i.e., the scene depth.
Specifically, a transmitter s transmits a pulse ps(t) to the scene. The pulse is reflected by a reflector at distance ds from the transmitter and received by a receiver at distance dr from the reflector, delayed by τsr=(ds+dr)/c, where c is the speed of the transmitted signals. Assuming the transmitter and the receiver are omnidirectional, the received signal is yr(t)=xps(t−τsr), where x is the reflectivity of the reflector.
Often, it is more convenient to express this delay in the frequency domain, i.e., Yr(ω)=xe−jωτsrPs(ω), where the uppercase denotes the Fourier transform. The propagation equation is linear, i.e., the principle of superposition can be used to describe the received signal from the transmission of multiple pulses and the reflection from multiple reflectors. Sensor directionality is straightforward to incorporate.
To describe a radar sensing system, the scene is considered in its entirety. The scene is discretized using a grid of N points and represent the reflectivity of each point using xn. The reflectivity is assumed constant as a function of frequency, although it is straightforward to model frequency-dependent reflectivity. Using τsrn to denote the propagation delay from transmitter s to receiver r through gridpoint n, the propagation equation becomes
                                          Y            r                    ⁡                      (            ω            )                          =                              Σ            n                    ⁢                      Σ            s                    ⁢                      x            n                    ⁢                      ⅇ                                          -                j                            ⁢                                                          ⁢                              ωτ                srn                                              ⁢                                                    P                s                            ⁡                              (                ω                )                                      .                                              (        1        )            
Model-Based Compressive Sensing
Compressive sensing enables significant improvements in the ability to acquire and reconstruct signals at a rate of their complexity rather than a rate of the ambient space in which the signal lies. This is achieved using a signal model. Conventional compressive sensing formulations assume the signal is sparse in some basis. The sparsity model, enforced during reconstruction, resolves the ambiguities in the underdetermined system arising from acquiring the signal at a rate lower than the ambient signal dimension.
As used herein, sparse is a well known term of art in signal processing, and not a relative indefinite term. A “sparse” approximation estimates a sparse vector—i.e., a vector with most coefficients equal or approximately equal to zero—satisfying a linear system of equations given high-dimensional observed data and an acquisition matrix.
Signal models other than sparsity can also be used to reduce the sampling requirements and improve the reconstruction performance. Manifolds, group sparsity, joint sparsity, and fusion frame sparsity models are example models. A large number of these models can be described by assuming the signal belongs to a union of subspaces, a more general model with well-known reconstruction processes and recovery conditions. Conventional signal sparsity is also a special case of the union of subspaces model.
Typically, a union of subspaces a is used in recovering a signal from measurements acquired using a linear systemy=Ax,  (2)where A describes the acquisition system, which is usually under-determined. The signal can be recovered, under certain conditions on. A, by determining a vector {circumflex over (x)} which belongs in the union of subspaces out of the ones that agree with explain the input signals.
Typical greedy processes, such as model-based Compressive Sampling Matching Pursuit (CoSaMP) and model-based Iterative Hard Thresholding (IHT) generalize their non-model-based counterparts, and attempt to optimize
                                          x            ^                    =                                    min              x                        ⁢                          ||                              y                -                Ax                            ⁢                              ||                2                                                    ,                              such            ⁢                                                  ⁢            that            ⁢                                                  ⁢            x                    ∈          S                ,                            (        3        )            where S is the space of signals admissible to the model. By replacing S with its convex relaxation, a convex optimization procedure can be used instead of the greedy processes.
Some greedy processes iterate between two basic steps. First, a candidate support for the signal of interest is identified, and second, the system over that support is inverted. Others iterate between improving the cost function ∥y−Ax∥2 and enforcing the restricted signal support. The model-based counterparts modify the support identification or enforcement step, in accordance to the support model for the signal of interest.
Model-based convex optimization procedures can be implemented in several ways. Usually the procedures attempt to balance the data fidelity cost ∥y−Ax∥2 with a model-based cost g(x)—a convex function penalizing deviations from the signal model. For example, in the standard sparsity model used in most compressed sensing applications, the model-based cost is the l1 penalty g(x)=∥x∥1. One approach to optimize such a convex cost is to alternate between a gradient descent improving the data fidelity part of the cost, and a hard or soft thresholding step improving the model-based cost.